How can you use trigonometric functions to simplify $4 e^{\frac{4 π}{3} i}$ $4 e^( ( 4 pi)/3 i )$ into a non-exponential complex number?

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How can you use trigonometric functions to simplify $4 e^{\frac{4 π}{3} i}$ $4 e^( ( 4 pi)/3 i )$ into a non-exponential complex number?

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Answer 1 · 2016-04-22T13:17:57

$- 2 - 3.464 i$ $-2-3.464i$

Explanation:

Euler's formula says that

$e^{i x} = cos x + i sin x$ $e^(ix)=cosx+isinx$ .

Inserting the value $x = \frac{4 π}{3}$ $x=(4pi)/3$ in the equation from the question and solve

$e^{\frac{4 π}{3} i} = cos (\frac{4 π}{3}) + i sin (\frac{4 π}{3})$ $e^((4pi)/3i)=cos((4pi)/3)+isin((4pi)/3)$
$= cos 240 + i sin 240$ $=cos240+isin240$
$= - 0.5 - 0.866 i$ $=-0.5-0.866i$

Multiply all of this by $4$ $4$ to give the final answer

$4 e^{\frac{4 π}{3} i} = - 2 - 3.464 i$ $4e^((4pi)/3i)=-2-3.464i$

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How can you use trigonometric functions to simplify $4 e^{\frac{4 π}{3} i}$ $4 e^( ( 4 pi)/3 i )$ into a non-exponential complex number?

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How can you use trigonometric functions to simplify $4 e^{\frac{4 π}{3} i}$ $4 e^( ( 4 pi)/3 i )$ into a non-exponential complex number?